Christoffel–Minkowski problem

E947537

mathematical problem problem in convex geometry

The Christoffel–Minkowski problem is a classical question in convex geometry that seeks to reconstruct a convex body from prescribed curvature or area measure data on the unit sphere.

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All labels observed (4)

Label	Occurrences
Minkowski problem	3
Aleksandrov problem	1
Christoffel problem	1
Christoffel–Minkowski problem canonical	1

How this entity was disambiguated

This entity first appeared as the object of triple T11812508 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.

NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07

Target entity: Christoffel–Minkowski problem
Context triple: [Elwin Bruno Christoffel, notableWork, Christoffel–Minkowski problem]

A. Nirenberg problem in differential geometry
The Nirenberg problem in differential geometry is a classical question about prescribing Gaussian curvature on the 2-sphere via conformal deformations of the metric.
B. Monge–Ampère equation
The Monge–Ampère equation is a fully nonlinear partial differential equation central to differential geometry, optimal transport, and several complex variables, often used to study curvature and geometric structures.
C. Yamabe problem
The Yamabe problem is a fundamental question in differential geometry concerning whether every compact Riemannian manifold admits a metric of constant scalar curvature within a given conformal class.
D. Minkowski’s theorem on convex sets
Minkowski’s theorem on convex sets is a fundamental result in convex geometry that characterizes lattice points in convex bodies, underpinning much of the theory of convex polytopes and the geometry of numbers.
E. Calabi conjecture
The Calabi conjecture is a fundamental result in complex differential geometry, proved by Shing-Tung Yau, which characterizes when a compact Kähler manifold admits a unique Ricci-flat Kähler metric in a given Kähler class.
F. None of above. chosen
G. Unsure - the case is ambiguous/there is not enough information to decide.

NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07

Target entity: Christoffel–Minkowski problem
Target entity description: The Christoffel–Minkowski problem is a classical question in convex geometry that seeks to reconstruct a convex body from prescribed curvature or area measure data on the unit sphere.

A. Nirenberg problem in differential geometry
The Nirenberg problem in differential geometry is a classical question about prescribing Gaussian curvature on the 2-sphere via conformal deformations of the metric.
B. Monge–Ampère equation
The Monge–Ampère equation is a fully nonlinear partial differential equation central to differential geometry, optimal transport, and several complex variables, often used to study curvature and geometric structures.
C. Yamabe problem
The Yamabe problem is a fundamental question in differential geometry concerning whether every compact Riemannian manifold admits a metric of constant scalar curvature within a given conformal class.
D. Minkowski’s theorem on convex sets
Minkowski’s theorem on convex sets is a fundamental result in convex geometry that characterizes lattice points in convex bodies, underpinning much of the theory of convex polytopes and the geometry of numbers.
E. Calabi conjecture
The Calabi conjecture is a fundamental result in complex differential geometry, proved by Shing-Tung Yau, which characterizes when a compact Kähler manifold admits a unique Ricci-flat Kähler metric in a given Kähler class.
F. None of above. chosen

Statements (44)

Predicate	Object
instanceOf	mathematical problem ⓘ problem in convex geometry ⓘ
appearsIn	modern convex geometric analysis literature ⓘ
appliesTo	convex bodies in Euclidean space ⓘ convex hypersurfaces ⓘ
asksFor	reconstruction of a convex body from area measure ⓘ reconstruction of a convex body from curvature data ⓘ
concerns	area measures on the sphere ⓘ convex bodies ⓘ curvature measures ⓘ
connectedTo	Aleksandrov–Fenchel inequalities ⓘ isoperimetric-type inequalities ⓘ
dimension	n-dimensional Euclidean space ⓘ
domain	unit sphere ⓘ
field	convex geometry ⓘ geometric analysis ⓘ partial differential equations ⓘ
generalizationOf	classical Minkowski problem ⓘ
goal	characterize which measures on the sphere arise from convex bodies ⓘ
hasAspect	existence of convex bodies with given curvature measure ⓘ regularity of solutions ⓘ uniqueness of convex bodies with given curvature measure ⓘ
hasCondition	balance conditions on the prescribed measure ⓘ positivity conditions on the prescribed measure ⓘ
involves	Gauss curvature ⓘ measure on the unit sphere ⓘ prescribed curvature on the unit sphere ⓘ support function of a convex body ⓘ surface area measure ⓘ
namedAfter	Elwin Bruno Christoffel ⓘ Hermann Minkowski ⓘ
relatedTo	Aleksandrov problem ⓘ Brunn–Minkowski theory ⓘ linked to: Brunn–Minkowski inequality Christoffel problem ⓘ linked to: Christoffel–Minkowski problem Minkowski problem ⓘ linked to: Christoffel–Minkowski problem Monge–Ampère equation ⓘ
status	partially solved in various dimensions and regularity classes ⓘ
studiedIn	differential geometry ⓘ measure theory ⓘ
typeOf	geometric reconstruction problem ⓘ inverse problem ⓘ
uses	curvature integrals ⓘ nonlinear elliptic PDEs ⓘ spherical measures ⓘ

How these facts were elicited